Bounds for the Utility-Indierene Pries of Non-traded Assets in
Inomplete Markets
D.G. Hobson
University of Bath
The author is supported by an
Epsr
Advaned Fellowship.
Marh 3, 2005
Abstrat
We onsider a speial lass of nanial models with both traded and non-traded assets and show
that the utility indierene (bid) prie of a ontingent laim on a non-traded asset is bounded above
by the expetation under the minimal martingale measure. This bound also represents the marginal
bid prie for the laim.
The key onlusion is that the bound and the marginal bid prie are independent of both the
utility funtion and initial wealth of the agent. Thus all utility maximising agents harge the same
marginal prie for the laim. This onlusion is in some sense the opposite onlusion to that of
Hubalek and Shahermayer (2001), who show that any prie is onsistent with some equivalent
martingale measure.
Keywords: Inomplete markets, non-traded assets, utility indierene prie, marginal prie, martingale measure.
JEL Classiation: G13.
AMS Subjet lassiation: 91B28, 91B16, 60J70.
Department of Mathematial Sienes, University of Bath, Bath, BA2 7AY, UK. dghmaths.bath.a.uk. Tel +44 1225
386989. Fax +44 1225 386492
1
Bounds for Utility Pries on Non-traded Assets
2
1 Introdution
The key insights of the Blak-Sholes option priing methodology are rstly, that in a omplete market
it is possible to repliate a ontingent laim, and seondly, that the initial fortune whih is required to
nane the repliating strategy is the fair prie of the option. In partiular, the prie of a ontingent laim
is determined unambiguously by the priniples of no-arbitrage, and is independent of the risk preferenes
of agents.
The onlusion that there are unique option pries in the Blak-Sholes model is lost as soon inompleteness is introdued into the model. This an happen in many ways, for example following the
introdution of transation osts, or if the assumptions of the model do not allow the agent to follow the
repliating strategy. In these ases any non-attainable ontingent laim arries risk, and any priing rule
makes impliit or expliit assumptions about utilities and preferenes.
The typial problem we have in mind (see Hubalek and Shahermayer (2001), Davis (1998), Henderson and Hobson (2002a, 2002b)) is as follows. There are two risky assets, one of whih is traded, but
the seond is not. Although the prie proesses for the assets may be driven by orrelated Brownian
motions, the oeÆients of the dynamis for the traded asset do not depend on the untraded asset. An
agent is due to reeive a laim whih is ontingent on the non-traded asset. How muh is that random
laim worth? This is the situation in real options, see Dixit and Pindyk (1994). An illustration from
Hubalek and Shahermayer (2001) is when the two assets are dierent brands of rude oil, only one of
whih is liquidly traded.
This problem is an example of the problem of priing a laim in an inomplete market, and is similar to
those onsidered in Follmer and Sondermann (1986), Follmer and Shweizer (1991), DuÆe and Rihardson
(1991) and many others. In ommon with DuÆe and Rihardson (1991) and Davis (1998) we model our
agents as maximisers of expeted utility. An alternative approah is to selet a martingale measure (for
example the minimal martingale measure) and to use that for priing.
The utility maximisation problem is a basi problem in nane and was rst studied in a ontinuous
time model by Merton (1969). A powerful approah to this problem is the dual variational method,
see, for example, Karatzas et al (1991), Kramkov and Shahermayer (1999) and Shahermayer (2001).
These papers provide a omplete solution of the optimal investment problem in an inomplete market.
The paper by Karatzas et al (1991), provides the foundations for both the notation and style of arguments
in this paper.
In order to address the question of the priing of ontingent laims in an inomplete market, Hodges
and Neuberger (1989) introdued the notion of the utility indierene prie. The utility indierene
bid prie is the amount the agent is prepared to pay whih leaves him indierent between paying for,
Bounds for Utility Pries on Non-traded Assets
3
and reeiving the laim, and not paying for, and not reeiving the laim. In order to derive this prie
we need to solve the utility maximisation problem both without the laim (see the referenes in the
previous paragraph) and in the ase with a random endowment. Cvitani at al (2001) have made some
steps towards a solution of the random endowment problem in a general setting. For the exponential
utility funtion, Delbaen et al (2002) haraterise the solution to the priing problem and determine the
assoiated dual problem.
The goal of this paper is to ompare the utility indierene prie aross dierent hoies of utility
funtion. Suh omparisons are very rare in the literature, although there have been some studies whih
investigate the impat of hanging the risk aversion within a parametri family of utility funtions, see
Sirar and Zariphopoulou (2005) (stohasti volatility models and exponential utility), Henderson et al
(2005) (stohasti volatility models and power-law utilities) and Bouhard et al (2001) (transations
osts and exponential utility). In general, the bid prie oered by an agent must depend on her hoie of
utility funtion, and there is a wide range of pries whih an be realised as the utility indierene prie.
However, in our spei non-traded asset setting, we show that there is a simple, non-trivial upper bound
on the bid prie for the option whih is independent of the hoie of utility. This bound is the prie of
the laim under the minimal martingale measure. Further, this bound represents the marginal prie, or
equivalently the unit prie she would be prepared to pay for an innitesimal quantity of the option.
The rest of this paper is strutured as follows. In the next setion we desribe the model and the main
onepts in an abstrat setting. We state the results, both purely in terms of probability, and in terms
of their nanial interpretation. In Setion 3 we prove the main theorems. The key observation is that
the bounds we derive are independent of the hoie of utility funtion. There is a set of analogous results
for ask pries whih we give in Setion 4. In Setion 5 we show how the non-traded assets model ts
into this framework and Setion 6 desribes the results for ertain ommon parametri families of utility
funtions for whih expliit alulations are sometimes possible. In Setion 7 we onsider a fundamentally
dierent model, whih is a speial ase of a stohasti volatility model, and whih also ts into the general
framework of Setion 2. Setion 8 onludes.
2 The main results and the assoiated nanial model
We suppose that we are given a probability spae (
; F ; P) with a xed -algebra G F , together
with a onvex funtion U : R ! [ 1; 1), whih is stritly inreasing, stritly onave and ontinuously
dierentiable on its domain, with derivative tending to zero at innity.
Assumption 2.1.
(a) Suppose that the probability spae supports a non-negative G -measurable random
4
Bounds for Utility Pries on Non-traded Assets
variable satisfying 0 < E [ ℄ < 1.
(b) Dene AG (x) = fX 2 mG : E [X ℄ xg, where mG is the set of G -measurable funtions, and let
AF be dened similarly, but with respet to the -algebra F . Suppose we are given an inreasing family
fA(x)gx2R, where A(x) is the set of admissible random variables for a given onstraint level x, with
AG (x) A(x) AF (x), and with the property that
A(x0 ) + (1 )A(x00 ) A(x0 + (1 )x00 )
8 2 (0; 1):
Let H be an element of mF + , the set of non-negative F -measurable random variables. We onsider
an optimal ontrol problem involving U and H . Set V (x) = supX 2A(x) E [U (X )℄. To avoid trivialities we
assume that V (x) < 1, for some, and then all x. Dene
V (x; k) =
so that V (x; 0) = V (x), and
sup E [U (X + kH )℄;
(1)
X 2A(x)
p(k) = inf fq : V (x q; k) V (x)g:
(2)
We now make a tehnial assumption, see also Karatzas et al (1991, Equation 6.2),
Assumption 2.2.
Suppose that for all w > 0 we have E [j (U 0 ) 1 (w )j℄ < 1.
Then, the main results of this paper are that, provided V (x ) >
1,
Theorem A.
p(k) kE [H ℄, and
Theorem B.
D+ pjk=0 = E [H ℄, where D+ denotes the right derivative.
In partiular, both the bound for p(k) and the derivative of p(k) near zero depend on the random
variable , but not on the funtion U , or the onstraint level x.
Let us now try to motivate these results by explaining why they are important and relevant in the
theory of mathematial nane.
Suppose that (
; F ; P) is a stohasti basis for a nanial market, where P is the real world probability
measure for an agent. The -algebra F represents all possible events in this model. We suppose that
there is also a sub--algebra G whih orresponds to the events assoiated with a omplete market whih
is embedded within the larger nanial model. The random variable plays the role of the (unique)
state-prie density in the omplete market model, and one of many state-prie densities in the larger
model.
5
Bounds for Utility Pries on Non-traded Assets
The analysis of this paper is based on this rather speial assumption that there exists the omplete
market model ontained within the larger nanial market. We do not laim that this assumption is
appropriate in a general nanial model, but rather that it is appropriate in ertain ontexts, and that
then some strong onlusions about the utility indierene priing of derivatives follow.
We fous on a single agent in this model who is assumed to have a onave utility funtion U . By
tradition the agent is female. She is assumed to be a maximiser of expeted utility of wealth. We assume
that the agent begins with initial wealth x and that the set of andidate or admissible target wealths for
the agent is the set A(x). It is natural to assume that A(x) inludes all terminal wealths whih an be
generated in the omplete sub-market (hene the assumption A(x) AG (x)), and onversely that every
admissible wealth must satisfy a budget onstraint relative to eah state-prie density in the inomplete
model. Let Z denote the set of state-prie densities in the inomplete model. Then we might dene
A(x) = A~F (x) = fX 2 mF : sup E [X ℄ xg;
2Z
and more generally other restritions on trading strategies may be imposed suh that A(x) A~F (x). In
either ase we have A(x) AF (x) = fX 2 mF : E [X ℄ xg.
Karatzas et al (1991) restrit the set of admissible random variables further to inlude only those
elements for whih E [U (X ) ℄ < 1. However, as we argue in Remark 3.6, this restrition is not neessary,
sine our assumptions guarantee that E [U (X )+ ℄ < 1 for all x 2 A(x), and hene E [U (X )℄ is always well
dened, even if it may equal minus innity.
The reent literature (Shahermayer (2001), Strasser (2004)) also ontains a disussion of the appropriate denition of admissibility, with speial referene to utility funtions dened on the real line. In a
dynami setting the budget onstraint is usually augmented by a further ondition whih ensures that
the disounted gains from trade is a supermartingale. However in our speial setting it turns out that
AG (x) A(x) AF (x) is both a suÆient and appropriate denition of admissibility, not least beause
we do not want to delare inadmissible to the agent operating in the full market any strategies whih
would normally be delared admissible in the omplete sub-market orresponding to G .
Our aim in this paper is to onsider the problem where the agent is to reeive k > 0 units of a random
non-negative payout H . We take the laim H as xed throughout. The agent's value funtion, now a
funtion of initial wealth and endowment k, is given by (1).
We want to deide how muh the agent is prepared to pay for the laim H . The utility indierene
(bid) prie (Hodges and Neuberger (1989)) is the amount of money the agent ould pay now whih would
leave her indierent between paying and reeiving the random laim H , and not paying, and not reeiving
the laim. Stated mathematially, if there is a unique q with V (x q; k) = V (x) then q = p(k) and we
Bounds for Utility Pries on Non-traded Assets
6
say that p(k) is the utility indierene prie. More generally, it may be that this quantity is not well
dened so we dene the bid prie p(k) for k units of the laim to be as in (2). We an also dene the
marginal bid prie for the agent to be D+ pjk=0 assuming this derivative exists. This denition is related
to the denition of the fair prie of a derivative given in Davis (1998). Davis denes the fair prie to be
Dpjk=0 , provided that D+ pjk=0 = D pjk=0 .
The existene of the marginal prie in a general inomplete market is the subjet of a reent paper by
Hugonnier et al (2005). For a xed utility dened on R+ , these authors are interested in the onditions
under whih a marginal prie exists and is unique (although it will depend on the hoie of utility).
Their denition of marginal prie orresponds to the fair prie of Davis. Loosely stated the result is that
the marginal prie is well dened for all bounded ontingent laims provided that the solution to the
dual problem denes an equivalent loal martingale measure. This later ondition plays a similar role to
Assumption 2.2.
The main results of this paper an be translated into the following statements. Under Assumptions 2.1
and 2.2,
Theorem Aa.
If h = E [H ℄ then kh is an upper bound on the bid prie for k units of the laim H , and
Theorem Ba.
The marginal bid prie for the non-negative laim H is given by h.
Note that h is independent of both the wealth of the agent and her partiular utility funtion. It is
also independent of the set Z of state-prie densities.
We an also show that kh is a lower bound on the ask prie for k units of the laim, where the ask
prie is dened in the natural fashion. If H is bounded then h is also the marginal ask prie.
The results of this setion have been desribed in a general setting, subjet to Assumption 2.1 on
the existene of a G -measurable state-prie-density, and the denition of admissible strategies. We now
desribe the type of situation where this assumption is satised. The key example we have in mind is a
model of non-traded assets. Suppose there are two risky assets given by orrelated (onstant parameter)
exponential Brownian motions. Suppose that only one of these assets is traded and onsider the problem of
trying to prie an option on the seond asset. The nanial sub-market onsisting of the traded asset alone
is a standard Blak-Sholes model, is omplete, and has a unique state-prie density. Assumption 2.1(a)
is satised in this example, and under some natural assumptions on the set of admissible strategies
Assumption 2.1(b) also holds. We return to this example in Setion 5.
7
Bounds for Utility Pries on Non-traded Assets
3 Proofs
The aim of this setion is to prove Theorems A and B under Assumptions 2.1 and 2.2. We begin by
stating some easy properties of the value funtion whih follow immediately from the properties of A and
the fat that U is onave.
(i) V (x; k) is inreasing in the rst argument. If
stritly inreasing.
(ii) V is onave in the (x; k) plane.
Lemma 3.1.
1 < V (x; k) < U (1) 1 then V is
Let z = inf fz : U (z ) > 1g and let y = D+ U (z ). For most ommonly used utility funtions
y = 1. Let the inverse to the derivative of U be denoted by I . The assumptions on U ensure that I is a
well-dened, ontinuous, stritly dereasing funtion on (0; y ). Let I (y) = I (y ) for y y if neessary.
Let x~ = x=E [ ℄. Note that X = x~ is an admissible element of A(x).
Lemma 3.2.
U (~x) >
1 if and only if V (x) > 1.
Clearly, V (x) U (~x). Conversely, if X 2 A(x) then P(X x~) > 0, and if U (~x) = 1 then
E [U (X )℄ = 1 and hene V (x) = 1.
The next result, adapted from Karatzas et al (1991), desribes the form of the optimal random variable
X^T . Note that if V (x ) > 1 then U (~x ) > 1.
Proof.
Suppose that V (x ) > 1. Then the optimal admissible element X^
X^ x I ( ), where is a Lagrange multiplier hosen to satisfy x E [I ( )℄.
Lemma 3.3.
X^ x
is given by
For w > 0 dene (w) = E [I (w )℄: Then (w) is a ontinuous, dereasing funtion, whih is
well dened by Assumption 2.2. On fw : (w) > z E [ ℄g the funtion is stritly dereasing, and we
an dene an inverse dened on (z E [ ℄; 1). It is simple to show that (1) = E [ ℄I (1) so that if
V (x ) > 1 or equivalently U (~x ) > 1 then (x) < 1. If we dene X^ I ( (x) ) then X^ 2 mG
and satises E [ X^ ℄ = ( (x)) = x, so that X^ 2 AG (x) A(x).
It remains to show that X^ is optimal. Observe that for any a and b, U (b) U (a) + (b a)U 0 (b). For
any X 2 A(x)
U (X^ ) U (X ) + (X^ X )U 0(X^ ) = U (X ) + (X^ X ) (x)
(3)
Proof.
and this last term has non-negative expetation under P. Hene E [U (X^ )℄ E [U (X )℄.
Corollary 3.4.
For z > 0, X^ x+z X^ x, almost surely.
Assumption 2.2 is used to show that the funtions and are well dened and hene that
X^ is optimal. This assumption an be weakened, see for example Kramkov and Shahermayer (1999),
Remark 3.5.
8
Bounds for Utility Pries on Non-traded Assets
where the notion of `reasonable asymptoti utility' is introdued and used to show that X^ = I ( ) is still
optimal.
In Karatzas et al (1991) the domain of admissible strategies is restrited to the lass for
whih E [U (X ) ℄ < 1. In fat this is not neessary. If X 2 A(x), then by denition E [(X )+ ℄ < 1, and
from (3), U (X ) U (X^ ) + (X X^ ) (x) . It follows that
Remark 3.6.
U (X )+ U (X^ )+ + (x)X + + (x) X^
and we onlude that E [U (X )+ ℄ is neessarily nite for all X 2 A(x), and E [U (X )℄ is well dened.
Now we an prove the rst main result.
Theorem 3.7.
Suppose
1 < V (x
). The quantity p(k) is bounded above by kh, where h = E [H ℄.
If h = 1 then there is nothing to prove.
Otherwise onsider x suh that V (x ) > 1. Let X^ x denote the optimal solution to the ontrol
problem when k = 0. Suppose z > kh and onsider the optimal ontrol problem (1) but with admissible
random variables onstrained to lie in A(x z ). Then, using U (b) U (a) + (b a)U 0 (a),
Proof.
E [U (X x z
+ kH )℄
E [U (X^ x )℄
E [(X x z
+ kH
=
E [(X x z
+ kH
X^ x)U 0 (X^ x )℄
X^ x) (x) ℄
(x)(x z + kE [H ℄
x) < 0
Optimising over admissible random variables we nd V (x z; k) < V (x): In partiular p(k) z . Sine
z > kh is arbitrary, it follows that p(k) kh.
Now we wish to investigate the way in whih p(k) depends on k. Clearly, if V (x ) > 1 and if
h < 1 then p(k) is nite. More generally, it is possible to show that provided V (x ) > 1, then p(k)
exits in [0; 1) as long as V (x; k) < U (1). For the rest of this setion we will be interested in p(k) when
k is small.
For k > 0, p(k) is non-dereasing and p(k)=k is non-inreasing. In partiular, if p(k)
exists in (0; 1) for any k, then 0 < p(k) < 1 for all k. Furthermore limk#0+ p(k)=k = (D+ p)jk=0 exists
in (0; 1℄.
Lemma 3.8.
The rst statement is obvious. For the statement about p(k)=k x 0 < k < k0 , and assume
p(k) < 1 and p(k0 ) > 0, else there is nothing to prove. Choose z > p(k) and u < p(k0 ). Then, by the
onavity of V
0
V (x z; k) < V (x; 0) kk0 V (x u; k0) + (k k0 k) V (x; 0) V (x ku=k0; k):
Proof.
9
Bounds for Utility Pries on Non-traded Assets
Hene z > ku=k0, and sine z and u are arbitrary it follows that p(k)=k p(k0 )=k0 .
The next result shows that if k is suÆiently small then p(k) is well dened.
If 1 < V (x ) V (x) < U (1) 1, and h = E [H ℄ < 1 then for suÆiently small k,
V (x p(k); k) = V (x).
Lemma 3.9.
Proof.
First note that for any X x 2 A(x),
E [U (X x
+ kH )℄ E [U (X^ x ) + (kH + X x X^ x)U 0 (X^ x)℄ = V (x) + (x)hk
Hene V (x; k) < U (1) for suÆiently small k.
Again for suÆiently small k, V (x 2kh) >
1 < V (x
1 and hene
2kh) < V (x 2kh; k) < V (x kh; k) V (x) < V (x; k) < U (1):
By Lemma 3.1 V (z; k) is onave and stritly inreasing in its rst argument for z 2 [x 2kh; x℄ so that
V (x q; k) = V (x) has a unique solution in the interval [x kh; x℄.
By Lemma 3.8 we know that D+ p exists at k = 0, and by Theorem 3.7 we know that (D+ p)jk=0 is
bounded above by h, and that this bound is independent of the onave funtion U and the level x. We
want to show that (D+ p)jk=0 = h.
Theorem 3.10.
Suppose
1 < V (x ) V (x) < U (1). Then
(D+ p)jk=0 = h = E [H ℄
(4)
By assumption U 0 is ontinuous, whih implies in turn that both I and are stritly dereasing,
at least over suitable ranges, and hene is ontinuous at x.
Suppose rst that H is bounded. Consider the optimal ontrol problem for xed k and X 2 A(x kz ).
By onavity of U , U (b) U (a) + (b a)U 0 (a) and for any admissible element
Proof.
U (X x kz + kH ) U (X^ x)
(X x
kz + kH
= (X x kz + kH
X^ x)U 0 (X^ x)
X^ x) (x)
Taking expetations, optimising over admissible random variables and dividing by k we nd
V (x kz; k) V (x) (x)fE [H ℄ z g:
k
0
Conversely U (b) U (a) + (b a)U (b) and X^ x kz is admissible, so
V (x kz; k) V (x) E [U (X^ x kz
+ kH )℄
(5)
E [U (X^ x )℄
+ kH X^ x)U 0 (X^ x kz + kH )℄
= E [(X^ x kz X^ x)U 0 (X^ x kz + kH )℄ + kE [HU 0 (X^ x kz + kH )℄:
E [(X^ x kz
(6)
10
Bounds for Utility Pries on Non-traded Assets
Now U 0 is a dereasing funtion so that U 0 (X^ x kz + kH ) U 0 (X^ x kz ) = (x kz ) . Note that
(x kz ) < 1 for suÆiently small k by the assumption that V (x ) > 1. Further, by Corollary 3.4
X^ x kz X^ x 0, so that the rst term in (6) is bounded below by (x kz )E [ (X^ x kz X^ x)℄ =
kz (x kz ). Now onsider the term E [HU 0 (X^ x kz + kH )℄. If H is bounded then by the ontinuity of
and the dominated onvergene theorem as k # 0 this onverges to (x)E [H ℄. Colleting together our
analysis of the two terms in (6) we dedue that
lim
k#0+
V (x zk; k) V (x) (x)fE [H ℄ z g:
k
Combining this result with (5) we have
lim
k#0+
V (x zk; k) V (x) = (x)fE [H ℄ z g:
k
(7)
By Lemma 3.9, for suÆiently small k, V (x) = V (x p(k); k) and so
p(k)
;1
k
rV = 0
provided the various quantities exist. By Lemma 3.8, (D+ p)jk=0 exists, and so, given (7), we dedue (4).
Now suppose that H is not bounded. Let Hn = H ^ n and let pn (k) denote the solution of (2) for k
units of the laim Hn . We have
D+ pjk=0 D+ pn jk=0 = E [Hn ℄ " E [H ℄:
But p(k) kE [H ℄ by Theorem 3.7 and so we dedue (4) for unbounded non-negative laims H .
The key result that makes the theorem work is the fat that is a ontinuous funtion.
In the proof this was ahieved by assuming that U had a ontinuous rst derivative. However, this is not
a neessary assumption and if the state prie density is a ontinuous random variable then may still
be ontinuous even if the derivative of U has jumps.
Remark 3.11.
4 Ask pries
In the previous setion we dened and proved results for the solution of (1) and (2) for positive k. In
eonomi terms these quantities give the bid prie for the laim H . Now we onsider negative k, whih
orrespond to ask pries.
Firstly observe that we an trivially extend the denition of V (x; k) given in (1) to inlude negative
k. Given this extension we an dene q(l) via q(l) = inf fq : V (x + q; l) V (x)g. Alternatively we an
set q(l) = p( l).
11
Bounds for Utility Pries on Non-traded Assets
Sine V is onave it is easy to dedue that q(l) p(l). Furthermore, the arguments leading to
Theorem 3.7 still hold, so that under our main assumptions we have that lh is a lower bound on q(l).
The orresponding results on the D+ qjk=0 D pjk=0 require additional assumptions on the random
variable H . For example, in Lemma 3.9 additional onditions are needed on H to ensure that q(l) is
nite. Essentially some ondition is needed to ensure that V (x; l) > 1. As we show in Example 6.1,
E [H ℄ < 1 is not suÆient. However, if H is bounded, then liml#0+ q (l)=l = D+ q jl=0 = E [H ℄: The
proof of this result proeeds almost exatly as in the proof of Theorem 3.10, apart from the fat that it
is not possible to extend from bounded laims to unbounded laims.
In the ase where the marginal bid and ask pries both exist and agree with eah other, then the
marginal prie is sometimes alled the Davis prie.
5 The non-traded assets problem
Having dened various onepts and stated the main results we now wish to desribe the fundamental
example that we have in mind. This example is a model with a non-traded asset, sometimes alled a
model with basis risk. In its simplest version the nanial market onsists of a bond with prie B growing
in deterministi fashion at rate r, and two further assets with pries S and Y desribed by exponential
Brownian motions. The dynamis are given by
dBt = Bt rdt;
dSt = St (dW + dt);
dYt = Yt (dZ + dt);
(8)
with B0 = b; S0 = s, and Y0 = y. The Brownian motions W and Z are orrelated with orrelation , and
p
we write dZ = dW + dW ? where = 1 2 and W ? is a Brownian motion whih is orthogonal to
W.
The problem faing our agent is to prie a ontingent laim on the non-traded asset Y , payable at
some xed time-horizon T , given that she is only able to hedge using the traded asset S and the bond B .
We need to desribe the state-prie densities and admissible trading strategies for the problem. We begin
by desribing the -algebras. Let F = (Ft )0tT be the ltration generated by the pair of Brownian
motions W and W ? , and let F = FT . Let G = (Gt )0tT and G = GT be generated by W alone.
The ltration G is generated by the asset S . In isolation the asset S and bond B form a Blak-Sholes
market. If the other perfet market assumptions of the Blak-Sholes model are satised then this market
is omplete and there are unique preferene independent pries for options on the asset S . Eah option
an be repliated by a dynami hedging strategy.
The asset Y plays the role of a non-traded asset. The fundamental problem is to prie a ontingent
laim on Y . If hedging on Y is allowed then the market is again omplete and there are unique preferene
12
Bounds for Utility Pries on Non-traded Assets
independent pries for all FT measurable laims. We are interested instead in the situation where hedging
using Y is not possible.
Sine Y is not traded there are many martingale measures for this problem. Let = ( r)= and
let
Z t
Z
1 t 2
1 2
?
t exp
du
dW
t = exp( rt) exp Wt
u u
2
2 0 u
0
R
for some Ft-adapted proess suh that 0T u2 du < 1, and suh that the nal exponential is a martingale. If we identify with T then Z = f g is the set of state-prie densities. Finally let = 0
(both as proesses and random variables on FT ). Then 2 mG + and Assumption 2.1(a) is satised. The
quantity plays the role of the unique state prie density in the redued omplete model (
; G ; P). Note
that t St is a P-martingale.
We all the minimal state prie density. This is beause is related to the minimal martingale
measure of Follmer and Shweizer (1991). The minimal martingale measure hanges the drifts of the
Brownian motions driving the traded assets to make the disounted pries of the traded assets into
martingales. It leaves unhanged the drifts of any Brownian motions whih are orthogonal to the Brownian
motions driving the traded assets. Hene in our setting Wt + t and Wt? are Brownian motions under the
minimal martingale measure. Note that under Assumption 2.1, is also the minimal entropy measure
and, more generally, the minimal distane martingale measure in the sense of Goll and Rushendor
(2001).
Now we onsider the spae of admissible strategies. In the absene of any ontingent laim the agent
seeks to maximise the expeted utility of wealth at time T . The spae AG (x) = fX 2 mGT : E [X ℄ xg
of attainable wealths satisfying the budget onstraint in the redued model an be rewritten as
AG (x) = fX : X = XT
C ; C 2 mGT+ ; Xt = t 1 x +
Z T
0
!
t dWt
; t Xt a martingale ; t 2 mGt g:
This is the spae of terminal wealths with an be generated using a dynami trading strategy involving
investments in S alone. (Note that dWt = d(t St )=( ), so that the integral in the above denition is
a disounted gains from trade.) The dynami strategy is adapted to the ltration generated by S , and
hosen suh that the disounted gains from trade are a martingale. This rules out doubling strategies.
At the nal time-point some wealth may be disarded.
It remains to speify the spae of admissible wealths A(x). The onept that we wish to represent is
the idea that an agent should not be allowed to trade on Y , but she should be allowed to use information
about the urrent value (and past history of Y ) in determining how muh to invest in the asset S . This
means that the integrand driving the gains from trade need not be adapted to Gt , but rather should be
13
Bounds for Utility Pries on Non-traded Assets
Ft -adapted. Given the representation of AG (x) it is natural to dene
A(x) = fX : X = XT C ; C 2 mF + ; Xt = t 1 x +
Z T
0
!
t dWt
; t Xt a martingale, t 2 mFt g:
With this representation it follows that Assumption 2.1(b) is satised.
It is also possible to view the model (8) as a restrition of a omplete market model in
whih both S and Y are traded assets. (The restrition is that the lass of strategies is redued from
investments in both risky assets to investments in S alone.) Under market ompletion (i.e. if trading in
Y is allowed) there is a unique state-prie density given by ^ where ^ = ( r)= and the unique fair
prie of a European ontingent laim with payo H (ST ; YT ) at T is given by
Remark 5.1.
E [H (ST ; YT )
^
℄:
There are no general relationships omparing the utility indierene prie p(k) (dened in the non-traded
assets model where Y is not traded) with kE [H (ST ; YT ) ^℄ (the omplete market prie when Y is traded).
For example, if H = YT then E [YT ^ ℄ y whereas
2
p(k)
= E [YT e rT e WT T=2 ℄ = ye T :
lim
k#0 k
Thus the marginal utility indierene prie in the non-traded assets model an be larger or smaller than
the unit prie under market ompletion, depending on the sign of . In ontrast, the main results of
this paper refer to the omplete market generated by S alone whih exists as a subset of the inomplete
model.
The denition of admissible wealths we give above is perfetly natural, and ideally suited
to providing the proofs of the main results, but is not the only denition of an admissible strategy used
in the literature on inomplete markets. Consider for example the notion of an aeptable strategy from
Delbaen and Shahermayer (1997). In the omplete market setting their denition is slightly more
restritive than the one used here, in that the spae of random variables C whih may be disarded at
time T is not neessarily the set of all positive random variables. (The reason for this is that Delbaen
and Shahermayer want to be able to onsider holding the same strategies long and short.)
Essentially the denition of an aeptable strategy in Delbaen and Shahermayer (1997) is that it
should generate a wealth proess whih dominates as a proess a maximal admissible strategy where
a maximal admissible strategy is a gains from trade proess whih is a true martingale under some
equivalent martingale measure. In ontrast, the denition we give above is related to the idea that our
admissible strategies dominate gains from trade proesses whih are true martingales under the minimal
martingale measure. Note that in the omplete market setting these two denitions are equivalent.
Remark 5.2.
14
Bounds for Utility Pries on Non-traded Assets
6 Examples
6.1
Exponential Utility
Consider the family of utility funtions with onstant absolute risk aversion. These utilities are dened
for negative wealth and take the form
U (x) = U (x) =
1 x
e ;
x 2 R;
with a positive parameter. For the dynamis given in Setion 5 we nd that the value funtion of the
agent is given by
2 T
1
rT
;
x 2 R:
exp xe
V (x) =
2
For this utility U 0 (x) = e x is ontinuous and tends to zero for large x. The inverse funtion I is given
by I (y) = (ln y)=. It follows that Assumption 2.2 is satised and (w) = ( 1=)E [ (ln w + ln )℄ =
e rT (ln w rT + 2 T=2)=. We nd (x) = exp( xerT + rT 2 T=2).
Suppose the laim takes the form g = g(YT ). Then the utility indierene bid prie for k units of the
laim is given in Henderson and Hobson (2002b) or Henderson (2002) as
p(k) =
1
ln E exp
(1 2 )
k(1 2 )g(YT ) :
(9)
Note that the prie does not depend on x sine wealth fators out of this problem. For g a non-negative
laim, and k > 0, it follows from Jensen's inequality that p(k) kE [g(YT )℄. Further, on dierentiation
we nd
D+pjk=0 = E [g(YT )℄:
Hene E [g(YT )℄, whih is independent of the risk aversion parameter , is both an upper bound on the
bid prie for the laim and the marginal bid prie.
Now suppose that we are interested in the ask prie. A formula for the ask prie is obtained by
replaing k with k in the right hand side of (9) and multiplying by 1. However if g(YT ) = YT then
the exponential moment is innite, and the utility indierene ask prie is also innite. Hene, trivially,
E [YT ℄ is a lower bound on the ask prie for the laim, but it does not represent the marginal ask prie.
6.2
Power Law Utility
For a positive risk aversion parameter R onsider the utility funtion
U (x) = UR (x) =
x1 R
;
1 R
x 0;
(10)
15
Bounds for Utility Pries on Non-traded Assets
with U (x) =
1 for negative x. The value funtion for the agent with no endowment is given by
x1 R e(1 R)rT
(1 R)2 T
V (x) =
;
exp
1 R
2R
with V (x) = 1 for x < 0.
The inverse to U 0 is given by I (y) = y 1=R and (w) = w 1=R E [ 1 (1=R) ℄ = w 1=R e(1 R)rT=R expf(1
R)2 =(2R2)g and for eah R, Assumption 2.2 holds.
In general there are no losed form expressions for options pries. However Henderson (2002, Theorem
4.2) gives an expansion for the prie in powers of the number of options bought. Suppose the wealth x
is positive. If the laim is units of the non-traded asset, then for k 0 the prie is given as
y2 2 R(1 2 ) 2( )=)T T
[e
p(k) = kE [YT ℄ k2 e rT
e
x
2
1℄ + O(k3 )
where = 2 2
R + R2 . For a more general laim g = g (YT ) the prie to leading order is p(k ) =
kE [g(YT )℄ + O(k2 ): In either ase the marginal bid prie is E [g(YT )℄ whih is independent of the risk
aversion parameter R. As for exponential utility the marginal ask prie an be innite for unbounded
laims.
Note that if we take R = 1 then we reover the formul for logarithmi utility.
2
6.3
A ounterexample for whih the bounds are not attained.
Consider an agent with power-law utility funtion given by (10) with R < 1. Suppose that this agent has
zero initial wealth and r = 0. We onsider the bid prie of this agent for k units of the laim YT where
YT = exp(ZT + ( 2 =2)T ), where for simpliity Z is independent of the Brownian motion driving
the traded asset.
In this ase V (0) = 0. If the agent bids any positive amount p for the laim then there is a positive
probability that the agent has negative nal wealth, and therefore her utility is 1. However, even if
the agent follows the strategy of investing zero in the traded asset, her value funtion is
E [U (k exp(BT
+ (
k1 R
2
=2)T ))℄ =
exp (1
1 R
R)T
R(1
2
2
R)
T > 0:
Hene there is no solution to the equation V ( p; k) = 0: For this problem the bid prie for the laim is
zero, as is the marginal bid prie, whih is not equal to h = E [YT ℄. The utility indierene prie is not
dened. However in this ase V (0 ) = 1 so that the hypotheses of the theorems are not satised.
16
Bounds for Utility Pries on Non-traded Assets
7 `Almost omplete' stohasti volatility models
In this setion we onsider a ompletely dierent situation to the main example of Setion 5. Here we
suppose that the triple (B; S; Y ) onsists of a bond, a traded asset, and a proess whih governs the
stohasti volatility of that asset.
Consider the model
dSt
= (Yt ; t)dW + (r + t (Yt ; t))dt;
dYt = (Yt ; t)dZ + (Yt ; t)dt; (11)
St
where, as before, dZt = dWt + ? dWt? . This is a standard stohasti volatility model, see, for example,
dBt = Bt rdt;
Hobson (1998) for a review and a list of popular parametri forms for , et. As written, the proess Y
driving the volatility is an autonomous diusion, but the results below remain valid even if ; ; and are arbitrary adapted funtions. The drift of the traded asset is (r + t (Yt ; t)) so we have parameterised
the dynamis of S in terms of the Sharpe ratio t .
Again S plays the role of a traded asset. This time Y is assoiated with the volatility of S rather
than being a seond nanial asset, but again Y is not traded. The state-prie densities for the model
take the form
T = e rT exp
Z T
0
t dWt
Z
!
1 T 2
dt exp
2 0 t
Z T
0
?
t dWt
Z
!
1 T 2
dt :
2 0 t
The model is inomplete and there are no unique preferene-independent option pries. Indeed Frey and
Sin (1999) show that in general there are no non-trivial bounds on the pries of all options in this model.
In other words the set of pries whih are onsistent with some risk neutral priing measure is the interval
(0; S0 ).
In general, in a stohasti volatility model there is no non-trivial utility-independent bound on the
utility indierene bid prie of an option. However, suppose that the Sharpe ratio is deterministi,
t = (t). (In the mathematial literature, a stohasti volatility model with this property is said to be
`almost omplete', see Pham et al (1998).) Then we an verify that Assumption 2.1(a) is satised, and
under appropriate modeling assumptions we an onlude that Theorems Aa and Ba hold. In partiular,
the marginal utility indierene bid prie for a ontingent laim H does not depend on the utility funtion
of the agent and is given by E [H ℄, where = T0 .
8 Conluding Remarks
In an inomplete market there are no unique, preferene independent option pries. Instead it is neessary
either to hoose a priing measure from the family of equivalent martingale measures, or to model the
Bounds for Utility Pries on Non-traded Assets
17
the preferenes diretly. Utility indierene option priing is a onsistent priing sheme whih redues
to Blak-Sholes option priing in a omplete market. Pries are non-linear, so that in general agents
will pay a lower unit prie for larger quantities of a nanial asset. These larger quantities are assoiated
with higher risk.
The pereived disadvantage of the utility based option priing approah is that sine the prie depends
on the hoie of utility funtion, it seems unlikely that there are any general priing priniples whih hold
uniformly aross all utility funtions, and for all initial wealths. In our non-traded asset model we have
shown that this is not the ase, and that there is a bound for the utility indierene bid and ask prie
for the option whih is valid for all suÆiently regular utility hoies. Furthermore, this bound also ats
as the marginal prie.
If the bid prie for every agent is below the ask prie for every agent then under no irumstanes
would any trading our. Hene if all agents have zero initial endowment of Y , and all agents have
the same physial measure P, then no trading in a laim with payo H (Y ) would our, even if the
agents have dierent initial wealths or utility funtions. Trading only ours if the agents have initial
endowments of non-traded asset, or if they have dierent expetations of Y .
The results of this paper should be ontrasted with the results of Hubalek and Shahermayer (2001).
They onsider the spei example we introdued in Setion 5, and onlude that for a all option on
the non-traded asset, every positive prie is onsistent with some equivalent martingale measure. The
reason for this is that although a hange of measure leaves the disounted prie of a traded asset as a
martingale, the drift on the non-traded asset is undetermined: there are martingale measures for whih
the non-traded asset has arbitrarily large drift (positive or negative). As a onsequene the expeted
payo of the all option an be made arbitrarily large or small with a judiious hoie of priing measure,
and no-arbitrage arguments give only trivial bounds on the prie of a ontingent laim.
We reah an opposite onlusion: for an agent who seeks to maximise expeted utility, the marginal bid
prie for a laim on the non-traded asset is uniquely speied. One reason why we get this uniqueness is
that we assume that the agent has zero initial endowment of the non-traded asset. If we were to relax this
assumption then the uniqueness would be lost. Note that there is no role in the no-arbitrage arguments
of Hubalek and Shahermayer for the agent's initial endowment of the non-traded asset, and hene they
must obtain wider bounds. However, when the assumption of zero initial endowment is appropriate, our
onlusion is very strong, and the only andidate for the marginal utility indierene prie of the laim
is the disounted expeted value under the minimal martingale measure.
Bounds for Utility Pries on Non-traded Assets
18
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